mean flow generation by g rtler vortices in a rotating annulus with librating side walls

Mean flow generation by Görtler vortices in a rotating annulus with librating
side walls
,
Abouzar Ghasemi V. , Marten Klein, Uwe Harlander, Michael V. Kurgansky, Eberhard Schaller,
and Andreas Will

Citation: Phys. Fluids 28, 056603 (2016); doi: 10.1063/1.4948406
View online: />View Table of Contents: />Published by the American Institute of Physics


PHYSICS OF FLUIDS 28, 056603 (2016)

Mean flow generation by Görtler vortices in a rotating
annulus with librating side walls
Abouzar Ghasemi V.,1,a) Marten Klein,1 Uwe Harlander,2
Michael V. Kurgansky,3 Eberhard Schaller,1 and Andreas Will1
1

Department of Environmental Meteorology, Brandenburg University of Technology
Cottbus–Senftenberg, Burger Chaussee 2, D-03044 Cottbus, Germany
2
Department of Aerodynamics and Fluid Mechanics, Brandenburg University of Technology
Cottbus–Senftenberg, Siemens-Halske-Ring 14, D-03046 Cottbus, Germany
3
A. M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevsky 3,
119017 Moscow, Russia

(Received 22 April 2015; accepted 4 April 2016; published online 11 May 2016)
Time periodic variation of the rotation rate of an annulus induces in supercritical
regime an unstable Stokes boundary layer over the cylinder side walls, generating
Görtler vortices in a portion of a libration cycle as a discrete event. Numerical results

show that these vortices propagate into the fluid bulk and generate an azimuthal
mean flow. Direct numerical simulations of the fluid flow in an annular container
with librating outer (inner) cylinder side wall and Reynolds-averaged Navier–Stokes
(RANS) equations as diagnostic equations are used to investigate generation mechanism of the retrograde (prograde) azimuthal mean flow in the bulk. First, we explain,
phenomenologically, how absolute angular momentum of the bulk flow is mixed
and changed due to the propagation of the Görtler vortices, causing a new vortex
of basin size. Then we investigate the RANS equations for intermediate time scale
of the development of the Görtler vortices and for long time scale of the order of
several libration periods. The former exhibits sign selection of the azimuthal mean
flow. Investigating the latter, we predict that the azimuthal mean flow is proportional
to the libration amplitude squared and to the inverse square root of the Ekman
number and libration frequency and then confirms this using the numerical data.
Additionally, presence of an upscale cascade of energy is shown, using the kinetic
energy budget of fluctuating flow. C 2016 Author(s). All article content, except where
otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
( [ />
I. INTRODUCTION

Understanding bulk mean flow generation mechanisms in rotating flows in weakly nonlinear
and weakly or intermittently unstable regimes is important, since they appear in a variety of
geophysical and technical bulk flows, and contribute to our understanding of turbulence.
It is well known that in a cylindrical container subjected to longitudinal libration, i.e., a sinusoidal modulation of the background rotation rate, an oscillatory Ekman boundary layer forms over
the top and bottom lids, and a Stokes boundary layer develops over the cylinder side wall. Nonlinear
effects in the oscillatory Ekman layer induce an azimuthal mean flow in the fluid bulk (Busse, 2010;
Sauret et al., 2012; and Noir et al., 2010). Busse (2010) using an analytical solution, Sauret et al.
(2012) performing 2D-numerical simulation using commercial software COMSOL Multiphysics,
and Noir et al. (2010) conducting a laboratory experiments using direct flow visualization with
kalliroscope particles, all illustrated the bulk mean flow driven due to the nonlinearities in the

a)

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© Author(s) 2016.


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oscillatory Ekman layer. Noir et al. (2009) found that for moderate libration amplitude in a weakly
nonlinear regime, the centrifugally unstable Stokes boundary layer becomes susceptible to Görtler
vortices. Nevertheless, the exact mechanism of mean flow generation by Görtler vortices or via the
centrifugally unstable Stokes boundary layer is still unclear.
From Rayleigh’s (1916) criterion we know that boundary layer over a curved wall can be
centrifugally unstable if square of angular momentum decreases with increasing radius. Görtler
(1955) showed theoretically that if the centrifugal force is large enough in comparison with the
viscous force, vortices can be generated in the unstable boundary layer (see also Saric, 1994). Floryan (1986) showed that the destabilization action of the centrifugal force over a concave (convex)
surface can produce secondary motions in the form of Görtler vortices. The effect of rotation on
the Görtler vortices is discussed theoretically by Zebib and Bottaro (1993) who considered a fluid
flow over a concave wall subjected to a uniform rotation around its axis of symmetry. They showed
that negative (in the opposite direction to the fluid flow velocity) rotation of a concave wall has
a stabilizing effect, whereas positive rotation has a destabilizing effect on the boundary layer flow
with respect to the formation of Görtler vortices. Chen and Lin (2002) obtained the same result as
Zebib and Bottaro (1993) for the concave surfaces and showed that the effect of rotation for the fluid
flow on a convex surface is opposite to that of a concave wall.

The above investigations considered steady boundary layer flow over fixed or rotating curved
walls. In contrast, in our setup inner (as a convex wall) and outer (as a concave wall) cylinder side
walls librate longitudinally, causing the Stokes boundary layer (Sauret et al., 2012 and Noir et al.,
2010). For librational frequencies up to a limit value, the Stokes boundary layer becomes centrifugally unstable—once the librational amplitude becomes supercritical—in the prograde (retrograde)
phase of a librational cycle over the inner (outer) cylinder side wall, thus generating Görtler
vortices.
Generation of a mean flow by centrifugal instability dates back to early work of Scorer (1965)
and Scorer (1966) who reported that if a fluid flow rotating about its axis is externally stirred on
small scales, a single large scale vortex may be generated with a mean swirl velocity uθ + Ωr
different from the solid body rotation velocity, Ωr (uθ is the azimuthal velocity in the co-rotating
frame of reference). Bretherton and Turner (1968) proposed the same argument and explained
how the presence of a preferred direction, arising from the Coriolis force, induces an anisotropic
mixing, resulting in a mean radial flux of angular momentum which homogenizes fluid’s angular
momentum (see also Manton, 1973). Thompson (1979) argued that when inertial wave amplitude
is large, centrifugal instability gives rise to a strong local mixing which produces a stable vortex
spreading along the axis of rotation (see also Maas, 2001). More recently, Kloosterziel et al. (2007)
investigated development of inertial instability in initially barotropic vortices in a uniformly rotating
and stratified fluid. They stated that centrifugally unstable vortices propagating beyond the unstable
region may mix angular momentum and produce a basin-scale vortex with a stable velocity profile
spreading along the axis of rotation, i.e., with homogenized angular momentum.
In this paper, we try to understand generation mechanism of the azimuthal mean flow caused
by the centrifugally unstable Görtler vortices. Zhang et al. (1997) investigated mean flow generation
by turbulence in a rotating annulus with a rough inner cylinder subjected to axial oscillation. They
used the Reynolds-averaged Navier–Stokes (RANS) equations as a diagnostic tool and showed that
generated turbulence by the oscillating rough inner cylinder induces a retrograde azimuthal mean
flow in its vicinity via the Reynolds stress terms. In contrast, in this study, we use the longitudinal
libration of a smooth either inner or outer cylinder side wall, producing a Stokes boundary layer.
For not too high libration amplitude, the Stokes boundary layer spawns Görtler vortices only in
a portion of each libration cycle as a discrete event. Instability develops and then decays before
the libration reaches the unstable phase of the subsequent cycle. Following the procedure used by

Zhang et al. (1997), here we try to illustrate azimuthal mean flow generation mechanism by the
Görtler vortices. In fact, this study is an attempt to answer the following question: do the Görtler
vortices induced by longitudinal libration of the cylinder side wall of a rotating annulus modify the
fluid flow in the bulk?
To answer this question, we conducted 3D direct numerical simulations (DNSs) of the fluid
flow in a rotating annulus with the cylinder side walls subjected to the longitudinal libration.


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Furthermore, a series of coarse-resolution numerical simulations have been performed to study the
scaling behavior of the mean flow and to clarify the relative importance of mean flows generated by
different mechanisms (i.e., nonlinearity in the oscillatory Ekman layer and Görtler vortices).
The paper is organized as follows: the governing equations, boundary conditions, numerical
method, and post-processing details are introduced in Section II. In Section III, we present the
numerical simulation results. First, we distinguish between the azimuthal mean flows generated by
the nonlinearities in the oscillatory Ekman layer and by the Görtler vortices, using the librational
boundary condition for the lids and/or cylinder side walls, and periodic and closed boundary conditions in axial direction. Evolution of the Görtler vortices is given in Sections III B and III C for inner
and outer cylinder libration cases. In Section III D, we discuss the transient RANS equations for the
time scale of the development of the Görtler vortices and the kinetic energy budget of turbulent flow
to elucidate the generation mechanism of the azimuthal mean flow. In Section III E, we consider the
long time-mean RANS equations and discuss the scaling behavior of the azimuthal mean flow with
respect to Ekman number, libration amplitude, and libration frequency.

II. GOVERNING EQUATIONS AND NUMERICAL MODEL
A. Governing equations

We consider a homogeneous and incompressible fluid with kinematic viscosity ν bounded
by two concentric cylinders and two end plates rotating about their common axis of symmetry,
i.e., Ω0 = Ω0ez (see Figure 1), where Ω0 is mean rotation rate of the annulus (in rad/s). In co-rotating
frame of reference governing equations take the well-known dimensionless form (Batchelor, 1967)
∂u
+ (u · ∇) u + 2ez × u = −∇P + E∇2u,
∂t
ν
∇ · u = 0, E =
,
Ω0 R22

(1)
(2)

where R2 is the outer cylinder radius and gives typical length scale and T = Ω−1
0 is the reference time
scale. Consequently, U = R2Ω0 and p0 = R22Ω20 are the velocity and pressure scales. Dimensionless
time is represented by t, dimensionless radius by r, dimensionless axial coordinate by z, dimensionless annulus height by H, and dimensionless inner and outer cylinder radii by r 1 and r 2, respectively.
Here we used two geometries for which H and r 2 are the same, and only radius of the inner cylinder
r 1 changes (Table I): r 1 = 1 for numerical simulations presented in Section III A, and r 1 = 1.5 for
the rest of the paper to save computational time. Schematic view of the geometry used in the current
paper is shown in Figure 1.

FIG. 1. Schematic drawing of the annular tank configuration. We use the classical Taylor–Couette geometry. Whole annulus
rotates with the constant angular velocity and additionally the boundaries (shown with colored lines) librate corresponding
to the defined boundary conditions in Table II. For geometry G1, r 1 = H = 1.5, and r 2 = 2, and for geometry G2, H = 1.5,
r 1 = 1, and r 2 = 2.0 (Table I).

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TABLE I. Details of the geometrical configurations.
Geometry

H

r1

r2

Section of the paper

G1
G2

1.5
1.5

1.5
1.0

2.0
2.0

III B–III E
III A

Wall libration is the only forcing mechanism present and in the laboratory frame of reference it
is defined as
Ω (t) = Ω0 [1 + ε sin (ωt)] ,

(3)

where ε is dimensionless libration amplitude (Rossby number) and ω is the dimensionless forcing
frequency. Libration of the cylinder side walls induces the Stokes boundary layer (Batchelor, 1967)
of dimensionless thickness

2E
δ=
.
ω
Given δ, we introduce a re-scaled radial coordinate
r − r1
r∗ =
.
δ
Furthermore uz , uθ , and ur stand for instantaneous velocity components in the axial, azimuthal, and
radial directions in the co-rotating frame of reference and r, θ, and z represent the corresponding coordinates. For the numerical simulations reported in the current work, we used librational
boundary conditions for either top and bottom lids or outer (inner) cylinder side wall or combinations of them (Table II). We used no-slip boundary condition, i.e., uz = ur = 0 at the annulus
walls. To impose librational boundary condition for the corresponding wall, the azimuthal velocity
uθ has been set according to (3) (Table II). Librating only the outer cylinder side wall, we also
used periodic boundary condition in axial direction, which is listed as boundary condition (V) in
Table II. Note that boundary condition (IV) in which top/bottom lids and outer cylinder side wall
are librating simultaneously is similar to the case investigated by Busse (2010), Wang (1970), Noir

et al. (2010), and Sauret et al. (2012).
B. Numerical method

Dimensionless form of governing Equations (1) and (2) is discretized after transformation
into generalized curvilinear coordinates to facilitate grid stretching at the walls. A second order
finite difference scheme is used in r and z directions and a Fourier spectral method in θ-direction.
Staggered contra-variant volume fluxes are model variables. For time integration, only wall-normal
viscous terms are treated implicitly using a factored Crank–Nicolson scheme which shows second
TABLE II. Librational boundary conditions for the azimuthal velocity u θ
in the co-rotating frame of reference. Boundary condition (IV) is similar to
the case investigated by Busse (2010), Noir et al. (2010), and Sauret et al.
(2012).
uθ
BC

r = r1

r = r2

z = 0, H

(I)
(II)
(III)
(IV)
(V)

0
ε r 1 sin(ωt)
0

0
0

ε r 2 sin(ωt)
0
0
ε r 2 sin(ωt)
ε r 2 sin(ωt)

0
0
ε r sin(ωt)
ε r sin(ωt)
Periodic


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order accuracy (Choi et al., 1993). Advection, Coriolis force, pressure gradient, viscous cross and
azimuthal derivatives, as well as boundary conditions are treated explicitly.
Time integration is done using a third-order Runge–Kutta scheme described by Orlandi (2000).
The “pressure-free” fractional step scheme of Kim and Moin (1985) is used to incorporate pressure and enforce incompressibility. The scheme conserves mass, energy, and momentum owing to
contravariant volume flux formulation in locally orthogonal grids (cf. Thompson et al., 1985 and
Morinishi et al., 1998). This numerical solver is called HYBRID-NEW solver. For more details on
the numerical method used in the HYBRID-NEW solver, the reader is directed to Klein et al. (2014)
and references therein.

Comparison between numerical results and laboratory experiment is reported by Klein et al.
(2014) and Borcia et al. (2014). Moreover, the solver is benchmarked against the study of Bilson
and Bremhorst (2007) who performed DNS of turbulent Taylor–Couette flow. Turbulence intensities
and off-diagonal
stress term were calculated. Results of the benchmark for the Reynolds
Reynolds

+
′
′
stress term ur uθ (superscript + denotes normalization by wall friction velocity) exhibited a relazθ
tive difference of about 2% (Figure 2(a)). The solver is also benchmarked for the case of libration
against the work of Sauret et al. (2012). They used commercial software COMSOL Multiphysics
to study mean flow generation by nonlinearities in an oscillatory Ekman layer in a rotating cylinder
subjected to the longitudinal libration. Comparison of time mean angular velocity (Ω2) (Figure 19
of Sauret et al., 2012; the case corresponds to ε = 0.05 and E = 4.0 × 10−5) revealed an absolute
difference of about 0.02 for the maximum amplitude of 2.64 (Figure 2(b)).
To resolve all the flow structures, we used DNS. Görtler vortices appear as longitudinal
counter-rotating rolls (or streaks) elongated in the azimuthal direction and distributed on the top of
one another in the axial direction (Noir et al., 2009 and Noir et al., 2010). This means that the flow
field under investigation is spatially anisotropic. Thus, the spatial resolutions in the azimuthal and
axial directions do not need to be as fine as in the radial direction.
In order to determine grid resolution needed in each coordinate direction for DNS, we followed
Moin (1995) and varied resolution in the axial, radial, and azimuthal directions independently and

FIG. 2. Comparison
of the HYBRID-NEW solver results with those of two previous studies. (a) shows Reynolds stress
+
term u ′r u θ′
(superscript + denotes normalization by wall friction velocity) obtained by Bilson and Bremhorst (2007,

zθ
Figure 27) and that obtained by the HYBRID-NEW solver for DNS of turbulent Taylor–Couette flow. (b) shows time mean
angular velocity (Ω2) obtained by Sauret et al. (2012, Figure 19(a)); the case corresponds to ε = 0.05 and E = 4.0 × 10−5 and
that obtained by the HYBRID-NEW solver for the mean flow generation by nonlinearities in the oscillatory Ekman layer.


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TABLE III. Details of the spatial discretization used for the results shown in different sections of the paper.

DNS
Coarse sim.
Coarse sim.

nr

nz

nθ

Ga

BCb

Paper section

∆r min

∆r max

∆z min

∆z max

401
401
451

201
201
201

256
65
65

G1
G1
G2

I, II
I
I, III, IV, V

III B–III D
III E

III A

0.001 03
0.001 03
0.002 8

0.0037
0.0037
0.0065

0.003 75
0.003 75
0.001 65

...
...
0.0045

NST c NEK d
12
12
4

...
...
4

a Geometry

(Table I).

boundary condition (Table II).
c Number of grid points resolving the Stokes boundary layer (thickness δ 0.0125).
dNumber of grid points resolving the oscillatory Ekman layer (thickness δ
EK 0.0075, Lopez and Marques, 2011).
bLibrational

performed a convergence study. Grid is refined in the radial direction near the librating wall using
tanh-stretching (Le, 1994). DNS is performed for the cases with boundary conditions (I) and (II)
(Table II). For these cases, the oscillatory Ekman layer does not form since the lids are fixed (in
the co-rotating frame of reference). According to our numerical results, another oscillatory Ekman
layer forms where the cylinder side wall meets the lids. However, its radial extent (≈δ) on the lids
is about two orders of magnitude smaller than the annular gap. Axial resolution study revealed that
its effect on statistics is less than 2%, excluding regions close to the lids. This means that its contribution to the azimuthal mean flow due to the Görtler vortices is negligible. Hence, an equidistance
grid is used in the axial direction. The grid is also equidistant in the periodic azimuthal direction.
Simulation details for the DNS and coarse numerical
are presented in Table III. Results

 simulations
′ u′
showed
an absolute difference of ≈3%
of the convergence study for Reynolds stress term u
r θ zθ
with extrapolated exact solution (Celik et al., 2008) for the DNS case. As can be seen in Section III
C (e.g., Figure 6), second order statistics approach zero at a distance ≈0.1 away from the librating
cylinder side wall. This ensures that second order statistics are not affected by the fixed cylinder side
wall.
The DNS (Sections III B–III D) and coarse numerical simulations (Section III A) were carried
out using dimensionless parameters ω = 0.514, ε = 0.6, and E = 4.0 × 10−5. Randomly perturbed
rigid body rotation state (perturbation amplitude uθ′ = 10−4 (cf. Marcus, 1984)) served as an initial

condition. Using staggered mesh (Arakawa C-grid), discontinuous boundary conditions at the corners, where the librating cylinder side walls meet the fixed lids, are avoided up to one half of a grid
cell; this is in contrast to a regularization method used in other schemes (Czarny et al., 2003).
C. Post-processing details

In order to investigate the mean flow generation mechanism, we introduce two types of averages in a statistically stationary state: (1) ensemble average for different phases of the libration cycle
to investigate the mean flow generation mechanism (Section III D), and (2) long time average over
several libration periods (due to the ergodicity, we found 10 libration periods to be enough), used in
Section III E.
We computed ensemble averages at selected phases (ϕ = ωt mod 2π) of a libration cycle to
investigate turbulence statistics using ergodicity hypothesis,
ψ¯ (ϕ) =

N
1 
ψ T1,ϕ + (n − 1) Tlib ,
N n=1

(4)

where T1,ϕ is the time of the selected libration phase within first libration period of the averaging,
Tlib is the libration cycle, and N is the number of the libration cycles over which the averaging is
performed.
Long time average is defined for an arbitrary function ψ (t) as
 t=T1+N Tlib
1
ψ˜ =
ψ (t) dt,
(5)
NTlib t=T1

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where ψ˜ denotes time averaging over a finite number of libration cycles, and T1 is the time at which
˜ Librating
the system reaches statistically stationary state. From (4) and (5), we obtain ψ¯ (ϕ) = ψ.
boundary can be considered as a “wave-maker” generating laminar velocity profile of the Stokes
boundary layer uθω (cf. Equation (8) in the work of Sauret et al., 2012 for an analytical velocity
profile of the Stokes boundary layer). Using this laminar profile, mean flow uθa (ϕ) generated in
unstable phase of the libration cycle via the centrifugal instability can be assessed,
uθa (ϕ) = u¯θ (ϕ) − uθω (ϕ) .

(6)

Since long time average of the Stokes boundary layer profile is zero, we obtain
u˜θa = u˜θ .
To compute the statistics, parts close to the top and bottom lids (z < 0.5 and z > 1.0) are
excluded. This minimizes any effect of the lids on the statistics. Thus the azimuthal mean flow can
be expected to be homogeneous in the azimuthal and axial directions due to symmetry property.
Due to homogeneity, statistics in the co-rotating frame of reference were computed by spatial averaging over the azimuthal direction (denoted by ⟨⟩θ ) and in some cases over the axial direction (denoted by ⟨⟩zθ ). This reduces computational
for stationary statistics. Spatial and time-averaged

 needs
′ u′
quantity is denoted by ⟨˜⟩θ or ⟨˜⟩zθ , e.g., u
.

r θ zθ
Using the long time and ensemble (phase) averages, two types of fluctuating velocity in the corotating frame of reference can be introduced. Using the long time average (used in
Section III E), decomposition of an arbitrary field ψ (r, θ, z,t) is defined as
ψ (r, θ, z,t) = ψ˜ (r, z)

θ

+ ψ ′ (r, θ, z,t) .

(7)

Using (7) for the instantaneous azimuthal velocity and adding and subtracting u¯θ (ϕ = ωt),
decomposition based on the ensemble averages is written as
uθ (r, θ, z,t) = ⟨u¯θ (r, z, ϕ)⟩θ + uθ′′ (r, θ, z,t) ,
where uθ′′ (r, θ, z,t)

(8)

is the fluctuation around u¯θ (ϕ) and is given by
uθ′′ (r, θ, z,t) = (u˜θ (r, z) − u¯θ (r, z, ϕ)) + uθ′ (r, θ, z,t) .

(9)

In a similar way, instantaneous radial and axial velocities can be decomposed based on the ensemble
averages.
Stationarity of the flow is investigated in terms of variation in time of kinetic energy of the
bulk flow (excluding a layer with thickness of 3δ); T1 is chosen as the beginning of the statistically
stationary state when the difference between kinetic energies corresponding to two consecutive
libration periods is smaller than 1%. Fulfilling this requirement for the axially periodic annulus
requires, due to the absence the top/bottom lids, a much longer simulation time than that of the

closed annulus (cf. Figure 3). In the axially closed annulus with either BC I or BC II, the azimuthal
mean flow caused by the Görtler vortices touches the lids and induces a weak classical Ekman
layer (not an oscillatory Ekman layer!); nevertheless, it does not affect statistics due to the flow
weakness. In contrast, this Ekman layer is absent for the axially periodic case. Formation of the
Ekman boundary layer for the axially closed annulus involves three different time scales, i.e., (i)
time span of development of the Ekman layer, (ii) spin-up time, and (iii) viscous diffusion time
scale (Greenspan and Howard, 1963). In contrast, the axially periodic case possesses just a single
(viscous diffusion) time scale due to absence of the Ekman layer. As a consequence, the azimuthal
mean flow penetrates deeper into the bulk for the axially periodic case (see Figure 3(b)).
III. RESULTS
A. Lid libration versus side wall libration

To determine influence of the boundary conditions listed in Table II on the generation of
azimuthal mean flow in the bulk, we discuss briefly the resulting azimuthal mean flows. We apply
boundary condition BC III to obtain an oscillatory Ekman boundary layer on the top and bottom lids. By applying boundary conditions BC I, BC II, and BC V, the Stokes boundary layer


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FIG. 3. Radial profiles of ⟨u˜ θ ⟩ zθ for simulations with the boundary conditions (Table II): outer cylinder in libration (I),
top/bottom lids in libration (III), top/bottom lids and outer cylinder in libration (IV), outer cylinder in libration and periodic
boundary condition in the axial direction (V), for ε = 0.1 (a) and ε = 0.6 (b), and E = 4.0 × 10−5, ω = 0.514. Note that the
geometry G2 is used for these simulations and curved profiles in (a) reflect the closeness of ω to resonance frequency of the
system.

is formed on the cylinder side walls. Libration of the top and bottom lids together with outer

cylinder side wall (BC IV) induces both the Stokes and oscillatory Ekman boundary layers. In this
section, we performed a series of coarse numerical simulations, using the geometry G2 (Table I).
Figure 3 shows radial profiles of the azimuthal mean flows generated using the librational boundary conditions BC I, III, IV, and V given in Table II. We used libration amplitudes ε = 0.1
(Figure 3(a)) and ε = 0.6 (Figure 3(b)) which correspond to the centrifugally stable and unstable Stokes boundary layers, respectively. Critical libration amplitude is found empirically to be
ε 0.13. For ε > 0.13, the Stokes boundary layer becomes centrifugally unstable and generates
the Görtler vortices (Figures 5 and 6). As Lopez and Marques (2011) showed, the vortices have
elongated or semi-symmetric structure in the azimuthal direction for the libration amplitude close
to the critical value (e.g., ε = 0.2), while for the larger libration amplitude (e.g., ε = 0.6), they are
strongly non-axisymmetric.
Using lid libration (BC III) with ε = 0.6 and 0.1, a retrograde azimuthal mean flow is found in
the bulk (Figures 3(a) and 3(b)), which is due to nonlinearities in the oscillatory Ekman boundary
layer (Noir et al., 2010). A prograde jet-like azimuthal mean flow is also found close to the outer
cylinder side wall for both ε (cf. Sauret et al., 2012).
Libration of the outer cylinder (BC I) with ε = 0.6 generates a retrograde azimuthal mean flow
in the bulk close to the outer cylinder side wall (Figure 3(b)). This mean flow is due to the Görtler
vortices. It has the same order of magnitude as that induced by BC III with ε = 0.6 (Figure 3(b)).
However, the stable Stokes boundary layer (ε = 0.1) induces an azimuthal mean flow which is
negligible in comparison with that induced by BC III with ε = 0.1 (Figure 3(a)).
Using outer cylinder libration boundary condition for axially periodic annulus (BC V) with
ε = 0.6, a retrograde azimuthal mean flow is generated by the Görtler vortices. The mean flow
occupies the whole bulk (Figure 3(b)). The weak classical Ekman layer (see last paragraph of
Section II C) is absent for the axially periodic annulus, causing the azimuthal mean flow to penetrate
more into the bulk compared with the closed annulus case (BC I).
Applying librational boundary condition for the top/bottom lids and outer cylinder side wall
(BC IV) with ε = 0.6, magnitude of the retrograde azimuthal mean flow in the bulk close to the
outer cylinder side wall is reduced compared with that of using BC I, while it is increased in the rest
of the bulk (Figure 3(b)). Using BC IV with ε = 0.1 corresponding to the stable Stokes boundary
layer, a retrograde azimuthal mean flow in the bulk and a prograde jet-like azimuthal mean flow
close to the outer cylinder side wall are excited, similar to BC III with ε = 0.1.
Considering the retrograde azimuthal mean flows induced by BC I, III, IV, and V with ε = 0.6

(Figure 3(b)), one can infer that the mean flow induced by the Görtler vortices affects the bulk flow
likewise the flow driven by the nonlinearities in the oscillatory Ekman boundary layer.


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As mentioned in Section II A, Noir et al. (2010) and Sauret et al. (2012) considered a librating cylinder which corresponds to BC IV in our study. They investigated azimuthal mean flow
due to the nonlinearities in the oscillatory Ekman boundary layer. Nevertheless, generation of the
azimuthal mean flow by libration of the cylinder side walls (BC I, BC II, and BC V), i.e., isolating
the effect of the centrifugally unstable Stokes boundary layer, has not been considered so far. In
the following, we consider either libration of the outer (BC I) or inner (BC II) cylinder side wall
(Table II) in a closed annulus and illustrate, by DNS means, generation mechanism of the azimuthal
mean flow driven by the Görtler vortices. Note that for the rest of the paper, the top and bottom lids
are fixed (in a co-rotating frame of reference) and the oscillatory Ekman boundary layer on them is
absent.
B. Flow evolution during a libration period

Spatial and temporal evolutions of the flow are shown in Figures 4(a), 4(b), and 4(e), respectively. Radial profiles of uθ at different phases (ϕ = ωt mod 2π) of a libration cycle for the inner
(Figure 4(a)) and outer (Figure 4(b)) cylinder libration cases (BC I and BC II, cf. Table II) show
spatial evolution of the flow. Azimuthal mean flow ⟨u˜θ ⟩zθ is also shown. The radial profiles cross
one of the Görtler vortices (indicated by a dashed horizontal line in Figures 5 and 7) and are
sampled at z 0.765 (z 0.725) for the outer (inner) cylinder libration case. The location selected
is not crucial due to the homogeneity of the azimuthal mean flow in the axial direction. As pointed
out by Sauret et al. (2012) and Noir et al. (2010), onset of the centrifugal instability is at the
peak of differential velocity, i.e., ϕ = π/2 (3π/2) when deceleration (acceleration) phase starts for
the inner (outer) cylinder libration case. In Figure 4(b) (Figure 4(a)), the radial profiles exhibit an

unstable state up to ∥r ∗ − r L B ∥ 5 (r L B: position of the librating cylinder side wall). Disturbed
part of the radial profile at ϕ = 25π/16 (Figure 4(b)) between r ∗ 36 and 37.5 shows head of the
vortex, which is resolved by approximately 14 grid cells. This suggests that a typical scale of the
Görtler vortices is on the order of the Stokes boundary layer thickness. For outer (inner) cylinder
libration case, ⟨u˜θ ⟩zθ reaches its maximum amplitude approximately at r ∗ 35 ( 4) outside the
Stokes boundary layer and decreases gradually to zero in the bulk. Interestingly, Figures 4(a) and
4(b) show that the sign of ⟨u˜θ ⟩zθ for the outer cylinder libration case is opposite to that of the inner
cylinder libration case. This will be discussed in more detail in Sec. III D.
Figures 4(c) and 4(d) show radial profiles of absolute angular momentum L corresponding to
Figures 4(a) and 4(b). Starting the simulation from the state of solid body rotation plus random
perturbation, negative (positive) angular momentum is released in the bulk during unstable phase
of a libration cycle, causing the bulk angular momentum to slightly decrease (increase) for the
outer (inner) cylinder libration case. This process occurs in each libration cycle. When statistically
stationary state is reached, viscous dissipation balances mean energy input by the Görtler vortices.
At this stage, final state of the mean angular momentum is established in the bulk and it does not
change anymore over the release of angular momentum in unstable phase of the libration cycle.
Mean angular momentum has a profile different from the solid body rotation as can be seen
in Figures 4(c) and 4(d). Values smaller (larger) than the angular momentum of the solid body
rotation (r 2) are interpreted as retrograde (prograde) mean flow. Since r 2 = 2r 1, effective libration
amplitude of the outer cylinder is two times that of the inner cylinder. In consequence, mean angular
momentum for the outer cylinder libration case is approximately twice that of the inner cylinder
libration case. Radial range of the mean angular momentum for the outer (inner) cylinder libration is
between r ∗ 35 ( 4), where ⟨u˜θ ⟩zθ is maximum, and r ∗ 20 ( 10).
Figure 4(e) shows time evolution of uθ for five libration periods for the outer cylinder libration
case at location (r, θ, z) (1.982, 1.5, 0.765) for ε = 0.6 and ε = 0.1 corresponding to the centrifugally unstable and stable Stokes boundary layers, respectively. For the case with ε = 0.1, time series
of uθ follows sinusoidal libration of the boundary. For the case with ε = 0.6, nonlinear motions
which are developed in a retrograde phase of the libration cycle become laminar before reaching
unstable phase of the next libration cycle. This introduces an asymmetry, resulting in a retrograde
mean flow. In addition, comparison of the evolution of uθ for five libration periods for ε = 0.6 shows
that axial position of the Görtler vortices is random.

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FIG. 4. Radial profiles of u θ at different phases during a libration period forthe inner (a) and outer (b) cylinder libration

cases and the corresponding angular momentum (c) and (d). ⟨u˜ θ ⟩ zθ and L
are shown as a gray line. Dimensionless
zθ

parameters are ε = 0.6, E = 4.0 × 10−5, and ω = 0.514. (e) Displays time series of u θ for 5 libration cycles for ε = 0.6 and
ε = 0.1 corresponding to the stable and unstable Stokes boundary layers, respectively.

C. Phenomenology of the evolution of instability and mean flow

In this section, propagation of the Görtler vortices into the bulk is discussed qualitatively.
Then two steps of instability development are presented: (i) generation of the Görtler vortices by
centrifugal instability, and (ii) mixing and redistribution of the bulk angular momentum.
The disturbed profiles in Figures 4(a) and 4(b) exhibited that the Görtler vortices propagate
into the bulk and decay there. To have a better understanding of the flow field within a Görtler
vortex on a phenomenological level, 2D fields of uθ and ur in (r, z)-plane are shown in Figures 5(a)
and 5(b). The 2D fields are shown for different libration phases corresponding to Figure 4(b)
and are sampled at θ 1.5. It can be seen that mushroom-like structure of the Görtler vortices
penetrates radially into the bulk and decays there. Moreover, ur is negative within the Görtler

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FIG. 5. Evolution of u θ (a) and u r (b) for the outer cylinder libration case in the unstable phase of a libration cycle. First
(from the right) vertical dashed line indicates border of the Stokes boundary layer (δ 0.0125), and the second line at r ∗ 3δ
shows where (θ, z)-cross sections in Figure 6 are sampled. Horizontal dashed line shows the axial coordinate at which the
radial profiles in Figure 4 are sampled. (c) Shows axial profile of u r at ϕ = 25π/16 sampled at δ < r 1.972 < 3δ. Triangles
in (c) show positions of cell center grid points. Dimensionless parameters are as in Figure 4.

vortices (see horizontal dashed line in Figure 5(b)), exhibiting motions into the bulk. In Figure 5(b),
negative values of ur are surrounded by positive values, implying overturning motions. Simply put,
it means that propagation of the Görtler vortices into the bulk is compensated by motions into
the Stokes boundary layer (see also Figure 5(c) for the motions amplitude). From columns 2-5 of
Figures 5(a) and 5(b), it can be understood that the vortices move radially with a velocity of order
ε (Figure 5(c)) which between ϕ = 25π/16 and 27π/16 can be characterized as an explosive radial
motion. For ϕ > 27π/16 mixing is dominating.


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FIG. 6. u θ (a) and u r (b) for the outer cylinder libration case at ϕ = 25π/16 sampled at r 1.962 (3δ away from the librating
outer cylinder side wall). Dashed line indicates the position at which (r, z)-cross sections in Figures 5 and 7 are sampled.
Dimensionless parameters are as in Figure 4.

Figures 6(a) and 6(b) show (θ, z)-cross sections of uθ and ur , respectively. The cross sections
are shown at ϕ = 25π/16 and sampled at r 1.962 which is about 3δ away from the librating outer
cylinder side wall. We see clearly that the Görtler vortices are azimuthally elongated vortex lines.
This implies an anisotropic structure of the flow field. Dashed line at θ 1.5 gives the position
of (r, z)-cross sections shown in Figure 5, which illuminates 3D spatial structure of the Görtler
vortices. Streaks of negative ur in Figure 6(b) are surrounded by positive values, exhibiting overturning motions. These overturning motions are similar to toroidal vortices of alternating sign in
the Taylor–Couette flow when the flow is unstable according to the Rayleigh criterion (Drazin and
Reid, 1981 and Kloosterziel et al., 2007, Figure 1). In the following, flow stability is discussed by
application of the Rayleigh criterion.
Rayleigh (1916) showed that in the absence of viscosity, an axisymmetric swirling flow is
unstable when magnitude of absolute angular momentum decreases with increasing radius in some
region of the flow. Rayleigh’s analysis has been applied by Kloosterziel and van Heijst (1991) to
inviscid, axisymmetric, homogenous flow in a co-rotating coordinate system. Modified Rayleigh
criterion in a co-rotating frame of reference reads
φ=

1 ∂L 2 2L 1 ∂L
= 2
= 2 (uθ /r + 1) (ξ z + 2) ≥ 0 stable, < 0 unstable,
r 3 ∂r
r r ∂r
∂(uθr)
L = uθr + r 2, and ξ z =
,
r∂r

(10)

where L is the dimensionless absolute angular momentum and ξ z is the dimensionless axial component of the relative vorticity. The Rayleigh discriminant φ is a scalar characterizing rotational or

centrifugal instability and exhibiting stable (φ > 0) and unstable (φ < 0) motions. Based on the
numerical results, gradient in the radial direction is much larger than that in the azimuthal direction,
even though the flow is 3-dimensional (Figures 6 and 5). This implies that ξ z ≈ ∂(uθr)/r∂r. Viscosity has a stabilization effect on the flow. However, the Rayleigh criterion has been derived based on
the assumption that the fluid is inviscid. Thus a part of the flow predicted to be unstable might be


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stable in practice. Hence we use φ to explain phenomenology of instability development, which can
be acceptable to a good approximation especially in the bulk (similar to Neitzel and Davis, 1981).
As can be inferred from (10), instability requires negative absolute vorticity (ξ z + 2) or negative
absolute velocity (uθ + r). Maximum velocity occurs at the boundary and since ε < 1, it is always
positive. Thus φ is negative if ξ z < −2. ξ z , φ, and radial component of relative vorticity (ξr ) in
(r, z)-plane are shown in Figure 7 for the outer cylinder libration case (BC I, Table II). Data are
sampled at θ 1.5 as indicated in Figure 6. Similar to ξ z , ξr ≈ −∂uθ /∂z, exhibiting variation of uθ
within the Görtler vortices in the axial direction; ξr illustrates again toroidal vortices of alternating
sign. From Figure 7, we see that stable and unstable regions are mixed (ϕ = 25π, 26π, 27π, and
31π/16), and then the flow becomes stable before reaching unstable phase of next libration cycle
(ϕ = π). Columns 1 to 4 of Figures 7(a) and 7(c) show generation of pairs of the centrifugally

FIG. 7. Evolution of ξ z (a), φ (b), and ξ r (c) for the outer cylinder libration case in the unstable phase of a libration cycle.
Dashed lines are as in Figure 5 and dimensionless parameters are as in Figure 4.


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unstable Görtler vortices, their propagation into the centrifugally stable bulk, and their decay. Furthermore, it is clear that outside the Stokes boundary layer wherever ξ z > −2, the flow is stable
(φ > 0) and vice versa.
Now considering (10) together with Figures 4(c) and 4(d), 5 and 7, it can be deduced that as
the Görtler vortices propagate into the bulk, uθ and thus L are changed outside the Stokes boundary
layer in such a way that ∂L/∂r is reduced. From (10), this reduces magnitude of φ as well. This is
in accordance with Bretherton and Turner (1968) who stated that stabilization process is due to the
mixing and transport of the angular momentum. This is also in agreement with the results presented
by Kloosterziel et al. (2007). They discussed that instability is stabilized via the mixing of angular
momentum in such a way that a new equilibrated vortex with a stable velocity profile emerges. This
equilibrated vortex is, in fact, a region with uniform or homogenized angular momentum (region
encompasses the azimuthal mean flow) discussed here.
Nonlinear interactions or mixing is expressed in terms of velocity correlations which appear
as the Reynolds stress terms in the momentum equations. Mixing can also be understood looking
at the vortices of alternating sign in ξr field (Figure 7(c)). Kloosterziel et al. (2007) hypothesized
that mixing of the vortices of alternating sign causes their propagation beyond the boundary of
unstable region (see their Figure 3). Here, this propagation can be characterized by high correlation
of negative uθ and negative ur (Figure 5), exhibiting nonlinear interactions.
Following Zhang et al. (1997), we think that the cause of the velocity correlation in our system
is the Coriolis force. A fluid parcel moving counter-clockwise (in the direction of background
rotation) experiences the Coriolis force ( f uθ′ ) which deflects the parcel in the positive radial direction, thus producing ur′ uθ′ > 0. Similarly, a fluid parcel moving clockwise (against the background
rotation) is deflected in the negative radial direction, which also yields ur′ uθ′ > 0. Although, locally,
production of ur′ uθ′ < 0 along the cylinder side wall cannot be ruled out, ur′ uθ′ > 0 is dominant due
to deflection of fluid particles by the Coriolis force. Thus,
we
 expect that averaging over time


> 0. In contrast to ur′ uθ′ , positive
or/and axial direction (due to ergodicity) yields always ur′ uθ′
zθ
and negative values of uz′ uθ′ are spatially distributed in the axial direction with approximately the
≈ 0. This unimodularity of ur′ uθ′
is shown in Sec. III D using the
same weight, causing uz′ uθ′
zθ
zθ
numerical data (Figures 8(a) and 8(b)).
In the previous paragraphs, we discussed the mixing or nonlinear interaction
of the vortices of

′
′
opposite sign, and their propagation into the bulk and the correlation term ur uθ . As we stated
zθ
already in the Introduction, Bretherton and Turner (1968) explained how the presence of a preferred
direction, caused by the Coriolis force, in the mixing process can induce an
mixing,
 anisotropic

resulting in a mean radial flux of angular momentum. The mean correlation ur′ uθ′
multiplied by
zθ
radius is identical to the mean radial flux of angular momentum (Bretherton and Turner, 1968 and
Zhang et al., 1997),





r ur′ uθ′
= r ur′ uθ′ + u˜θ + rΩ0
.
zθ

zθ



shows mixing and radial flux of the angular
As a result, the Reynolds stress term ur′ uθ′
zθ
momentum. Hence, one can say that the mixing is essential for the radial transfer of angular
momentum.
As a summary of the discussion above, it can be concluded that the unstable Stokes boundary
layer breaks up into the small scale Görtler vortices in (r, z)-plane which propagate into the bulk due
to the Coriolis force and change the angular momentum due to the action of Reynolds stresses. This
leads to a basin-scale vortex (azimuthal mean flow seen in the co-rotating frame of reference) at the
expense of small scale Görtler vortices.
D. Mean flow generation mechanism

In this section, we use the RANS equations in a co-rotating coordinate system to identify terms
responsible for the generation of azimuthal mean flow outside the Stokes boundary layer. Terms of
the RANS equations are computed using 3D DNS data for the inner and outer cylinder libration


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

FIG. 8. Radial profiles of the axially and azimuthally averaged Reynolds stress term u ′r u θ′
at different phases after
zθ
the onset of instability for the inner (a) and outer (b) cylinder libration cases. (c) shows time series of u ′r u θ′ at (r, θ, z)
(1.982, 1.5, 0.765), during 5 libration periods, and is depicted similar to Figure 4(e). Dimensionless parameters are as in
Figure 4.

cases. To avoid the lids effect, part of the annulus around middle-plane is considered to compute
statistics, i.e., 0.5 < z < 1.0. Instantaneous velocity is decomposed according to (8) into the phase
average u¯ x and fluctuating parts u ′′x in x = (r, θ, z) directions. Due to axial symmetry of the system
dominating mean flow is u¯θ . We assume statistical homogeneity in the θ and z-directions and use
ergodicity hypothesis to reduce computational demands by investigating azimuthally and axially
averaged quantities, i.e., ⟨u x ⟩zθ . θ-component of the RANS equations in the cylindrical coordinates
in the co-rotating frame of reference reads
∂⟨u¯θ ⟩zθ
∂⟨u¯θ ⟩zθ ⟨u¯r ⟩zθ ⟨u¯θ ⟩zθ
+ ⟨u¯r ⟩zθ
+
+ f ⟨u¯r ⟩zθ
∂t
∂r
r
( 2
)
∂ ⟨u¯θ ⟩zθ 1 ∂⟨u¯θ ⟩zθ ⟨u¯θ ⟩zθ

=ν
+
−
(11)
r ∂r
∂r 2
r2




∂ ur′′uθ′′
ur′′uθ′′
zθ
zθ
−
−2
,
∂r
r
where f = 2 is the Coriolis parameter.
Since the dominating mean flow is ⟨u¯θ ⟩zθ , we assume that ⟨u¯θ ⟩zθ ≫ ⟨u¯r ⟩zθ ≈ ⟨u¯ z ⟩zθ ≈ 0. Hence,
only time-dependent term ∂⟨u¯θ ⟩zθ /∂t remains on the left hand side of (11). To further simplify
this equation, we consider a region which starts from edge of the Stokes boundary
layer and ends

where amplitude of ⟨u˜θ ⟩zθ becomes maximum (Figures 4(a) and 4(b)) or ur′ uθ′
0 (Figures 8(a)
zθ
and 8(b)). As shown in Figure 4, a typical spatial scale of the radial disturbance is about a thickness of the Stokes boundary layer which is much smaller than the domain size. This implies that

curvature terms in the right hand side of (11) outside the Stokes boundary layer can be neglected
in comparison with the radial gradient terms (i.e., ∂u ′x /∂r ≫ u ′x /r ) (see also Zhang et al., 1997).


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



′ ′
′ ′

A comparison of Figures 10 and 12(b) confirms that ∂ u
r uθ zθ /∂r ≫ ur uθ zθ which is valid for


ur′′uθ′′ as well. In a development stage of the Görtler vortices (∆ϕ π/4 for the outer cylinder
zθ
libration case, see Figure 5), viscous terms outside the Stokes boundary layer can be neglected in
comparison with the Reynolds stress terms. Assuming that fluctuating radial velocity obtained by
the phase and long time averages to be equal ( ur′′ (r,t) zθ ≈ ur′ (r,t) zθ which is confirmed by the





numerical results) and using (9), one obtains ur′′uθ′′ ≈ ur′ uθ′ . Finally, the resulting equation for
zθ
zθ
development stage of the Görtler vortices reads


∂ ur′ uθ′
∂⟨u¯θ ⟩zθ
zθ
∝ −
.
(12)
∂t
∂r
Relation (12) describes a mechanism which drives the mean flow and renders an upscale cascade
of kinetic energy: from the Görtler vortices having spatial extent δ to the
mean flow
 basin-scale

having an extent of r 2 − r 1 = ∆r. According to (12), radial distribution of ur′ uθ′
governs the sign
zθ
and
 magnitude

 ofthe azimuthal mean flow. It is worth noting that numerical

 results revealed that
≫ uz′ uθ′
which, a posteriori, justifies neglecting the term uz′ uθ′

in (12). As stated
ur′ uθ′
zθ
zθ
zθ
′ ′
before, the fact
that
positive
and
negative
values
of
u
u
are
distributed
in
space
with the same
z θ


0.
weight causes uz′ uθ′
zθ 

Radial profiles of ur′ uθ′
at different libration phases ϕ and time series of ur′ uθ′ are shown
zθ

in Figure 8. Time span and sampling location of time series of ur′ uθ′ are similar to Figure 4(c). In
addition to the radial flux of angular momentum, ur′ uθ′
shows mean turbulent radial transport of
zθ


azimuthal momentum; ur′ uθ′
is always positive and its maximum appears close to the boundary
zθ


layer. This confirms the discussion presented in Sec. III C that ur′ uθ′
is always positive due to
zθ


deflection of the fluid particles by the Coriolis force. Figures 8(a) and 8(b) exhibit that ∂ ur′ uθ′ /∂r
zθ
outside the Stokes boundary layer is positive (negative) for the case of outer (inner) cylinder libration. Accordingly, relation (12) gives the sign of ⟨u¯θ ⟩zθ , which is negative (positive) for the case of
outer (inner) cylinder libration.


reaches its peak value at ∆ϕ = 3π/16 = 11π/16 − π/2 after the
It is worth noting that ur′ uθ′
zθ
onset of instability for the case of inner cylinder libration (Figure 8(a)) and ∆ϕ = π/16 = 25π/16 −
3π/2 for the case of outer cylinder libration (Figure 8(b)). This is consistent with faster development
of rotational instability for the case of outer cylinder libration due
absolute velocities.


 to higher
< 0 in the vicinity of axially
Using (12), Zhang et al. (1997) confirmed ⟨u˜θ ⟩zθ < 0 due to ur′ uθ′
zθ
oscillating rough circular shaft which can be regarded as an inner cylinder with small diameter. This
is on the contrary to our findings about the mean flow’s direction in vicinity of the inner cylinder,
i.e., ⟨u˜θ ⟩zθ > 0. To our understanding, this difference arises from the type of oscillatory forcing which
causes the Reynolds stress term to have inputs of different phases for the azimuthal oscillation (longitudinal libration) compared with axial oscillation.
Generation of the azimuthal mean flow can also be understood by investigating the kinetic energy
budget of mean and fluctuating flows. Production term appears with opposite signs in the two budget
equations. Interaction between strain rate of the mean field and turbulent stress yields production
of turbulence kinetic energy (Davidson, 2013). Kinetic energy budget of the fluctuating flow reads
(Davidson, 2013 and Moser and Moin, 1984)
∂ K¯
= P¯ + C¯ + T D + PD + PS + V D + ϵ,
¯
∂t

(13)

where K¯ is the total kinetic energy of the fluctuating flow; P¯ is the production term, C¯ the convection
or bulk velocity gradient production term, T D the turbulent diffusion, PD the pressure diffusion, PS
the pressure strain, V D the molecular diffusion, and ϵ¯ the dissipation term. Turbulent kinetic energy
budget equation can be obtained by multiplying the fluctuating momentum equations by the fluctuating velocity. Variation in time of the total kinetic energy is controlled by P¯ and ϵ;
¯ in their turn P¯


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and ϵ of the ensemble averaged equation are defined as (see also Moser and Moin, 1984)
∂⟨u˜r ⟩θ ur′ uθ′
∂⟨u˜r ⟩θ
⟨u˜θ ⟩θ − uz′ ur′
P¯ = P¯r r + P¯θθ + P¯z z , P¯r r = −ur′ ur′
+
,
∂r
r
∂z
∂⟨u˜ z ⟩θ
∂⟨u˜ z ⟩θ
∂⟨u˜θ ⟩θ uθ′ uθ′
∂⟨u˜θ ⟩θ ¯
⟨u˜r ⟩θ − uz′ uθ′
P¯θθ = −ur′ uθ′
−
, Pz z = −uz′ ur′
− uz′ uz′
,
∂r
r
∂z
∂r
∂z
(
)

(
)
∂u ′ ∂u ′ ∂u ′ ∂u ′
∂ur′
1 ∂ur′
ϵ = ϵ r r + ϵ θθ + ϵ z z , ϵ r r = 2E − r r − r r − 2
− uθ′
− uθ′ ,
∂z ∂z
∂r ∂r
∂θ
r ∂θ
(
)( ′
)
∂u ′ ∂u ′ ∂u ′ ∂u ′
∂uθ
1 ∂uθ′
ϵ θθ = 2E − θ θ − θ θ − 2
+ ur′
+ ur′ ,
∂z ∂z
∂r ∂r
∂θ
r ∂θ
(
)
∂uz′ ∂uz′ ∂uz′ ∂uz′
1 ∂uz′ ∂uz′
.

ϵ z z = 2E −
−
− 2
∂z ∂z
∂r ∂r
r ∂θ ∂θ

(14)

(15)

Taking into account ⟨u˜r ⟩θ ≈ ⟨u˜ z ⟩θ ≈ 0 and neglecting curvature terms, the only production term
−ur′ uθ′ ∂⟨u˜θ ⟩θ /∂r remains. After averaging in the homogenous z and θ-directions, P¯ reads
P¯θθ

zθ

 ∂⟨u˜θ ⟩zθ

≈ − ur′ uθ′
.
zθ
∂r

Sign of P¯θθ zθ determines presence of direct (downscale) or inverse (upscale) cascade of energy.
Figure 9(a) shows radial profiles of the production P¯θθ zθ at different libration phases after the onset
of instability for the outer cylinder libration case; positive P¯θθ zθ -values exhibit inverse (upscale)
cascade of energy. The same result
is


 found for the inner cylinder libration case due to the sign reversal
of ∂⟨u˜θ ⟩zθ /∂r (Figure 4) and ur′ uθ′
(Figure 8). Thus, libration of the inner and outer cylinder side
zθ
walls leads to an upscale cascade of energy in the unstable regime. Figure 9(b) shows dissipation rate
of the turbulent kinetic energy ⟨ϵ⟩zθ for the same phases as for the production term. ⟨ϵ⟩zθ has a peak
at the librating boundary and decreases to zero in the bulk.
E. Azimuthal mean flow in the steady state

In this section, we consider long time-mean RANS equation to find scaling behavior of the
azimuthal mean for the outer cylinder libration case. Long time averaging cancels out the time
dependent term ∂⟨u˜θ ⟩zθ /∂t. Hence mean forcing term is balanced by mean friction. Equation (16) is
derived in the same way as (12). Neglecting the mean flow and its derivatives in the radial and axial
directions outside the Stokes boundary layer (u˜r ≈ u˜ z ≪ u˜θ , ∂z ≈ ∂θ ≪ ∂r ) and mean fluctuations

FIG. 9. Radial profiles of P¯ θθ zθ and ⟨ϵ⟩zθ for the outer cylinder libration case at different phases after the onset of
instability. Dimensionless parameters are as in Figure 4.


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(u˜r′ ≈ u˜ z′ ≈ u˜θ′ ≈ 0), we obtain


′ ′
∂ u

r uθ

∂ 2⟨u˜θ ⟩zθ
.
(16)
∂r
∂r 2
Left and right hand sides of (16) are radial turbulent transport (RT) and dissipation (D) terms, respectively. Similar to Sec. III D, we considered a region which starts from the edge of the Stokes
 boundary

layer and ends where amplitude of ⟨u˜θ ⟩zθ becomes maximum (Figures 4(a) and 4(b)) or ur′ uθ′
≈0
zθ
(Figures 8(a) and 8(b)). Part of the annulus around middle-plane, i.e., 0.5 < z < 1.0, is considered
to compute statistics. As we already stated, the disturbed part of the radial profile outside the Stokes
boundary layer (Figure 4) shows a typical√length scale of the Görtler vortices to be on the order of δ.
From (16) and by considering ∂r ≈ δ ∝ E/ω, we obtain
zθ

≈E



1
′ ′
⟨u˜θ ⟩zθ ∝ u
r uθ zθ √
Eω

(17)

which predicts a scaling behavior of the azimuthal mean flow with respect to the dimensionless
parameters E and ω. We assume a linear dependency of the fluctuating velocities on ε (ur′ ∝ uθ′ ∝ ε),
resulting in ur′ uθ′ ≈ ε 2. This linear dependency is obvious for uθ′ and ur′ which can be understood from
Figure 5(c). Thus, from (17) we obtain a scaling ⟨u˜θ ⟩zθ ∝ ε 2. In order to validate proportionality (17)
we performed a series of coarse numerical simulations (Table III) and explored dependency of the
azimuthal mean flow on the dimensionless parameters E, ω, and ε.
Figures 10(a) and 10(b) show radial profiles of RT and D given in (16) for an axially periodic
and closed annulus, respectively. In the former case, the difference between RT and D is less than
1%. This confirms that governing dynamics depends mainly on these two terms.
A 30% difference between RT and D in (16) is found for the axially closed annulus (see
Figure 10(b)). It seems that the results are affected by the lids since the mean radial and axial velocities
are not precisely zero, however still smaller than the mean azimuthal velocity. An analysis of the
terms neglected in (16) showed that two more terms are needed to be considered. We obtain




′ ′
∂ u
r uθ zθ
∂ 2⟨u˜θ ⟩zθ
∂ u˜θ
.
(18)
+ f ⟨u˜r ⟩zθ ≈ E
+ u˜r
∂r
∂r zθ
∂r 2

√
Considering ∂r ≈ δ ∝ E/ω, we obtain
√

1

) f
′ ′
u
u˜r ∝ u˜θ .
r uθ + u˜θ u˜r +
ω
Eω
(

(19)

FIG. 10. Radial profiles of the terms in Equation (16) for the axially periodic annulus (a), and the terms of Equation (18) for
the axially closed annulus (b). The thick solid line in (b) shows the difference between the left and right hand sides of (18).
The simulations are performed for the case of outer cylinder libration. Dimensionless parameters are as in Figure 4.


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Ghasemi V. et al.

FIG. 11. Dependency of
ω = 0.514. Best fit gives



′ u′
u
r θ



Phys. Fluids 28, 056603 (2016)

(•) and ⟨u˜ θ ⟩ zθ

zθ max
0.885ε 1.85 (solid

line) and

max ( ) on the libration
0.039ε 1.86 (dashed line).

amplitude ε for fixed E = 4 × 10−5 and

Figure 10(b) shows Coriolis force (L A1 = f ⟨u˜r ⟩zθ ) and mean radial transport (L A2 =
⟨u˜r ∂ u˜θ /∂r⟩zθ ) in (18). Difference between the left and right hand sides of (18) is less than approximately 10% for the closed annulus.
Figure 11 shows maximum azimuthal mean velocity ( ⟨u˜θ ⟩zθ max) and maximum Reynolds stress


′ ′
) for ε ∈ [0.2, 0.6], E = 4 × 10−5, and ω = 0.514 for the axially closed annulus.
term ( u
ru

θ zθ
max

Both trend lines in Figure 11 exhibit ≈ε 1.85-scaling. This confirms the theoretical ε 2-scaling for
⟨u˜θ ⟩zθ max with approximately 10% deviation in the range 0.2 < ε < 0.6 investigated. This deviation
might be due to effect of the lids, nonlinearity of the regime investigated, and/or conducting coarse
numerical simulations.


′ ′
Radial profiles of ⟨u˜θ ⟩zθ (scaled by ε 2) and u
r uθ zθ in Figure 12 exhibit a constant radial position


∗
′ ′
r max
of ⟨u˜θ ⟩zθ max and u
. This implies that the Stokes boundary layer is invariant
r uθ
zθ max

∗
≈ 5δ can be regarded as an effective penetration
depth of the
with respect to ε. Radial position r max


′ u′
in

the bulk is
Görtler vortices in Figure 12. Furthermore, we see clearly that a slope of u
r θ zθ
increased with increasing ε. As given in (16), ⟨u˜θ ⟩zθ increases accordingly.



−5 and
′ u′
FIG. 12. Radial profiles of ⟨u˜ θ ⟩zθ /ε 2 and u
r θ zθ for different libration amplitudes ε, for fixed E = 4 × 10
ω = 0.514.


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Ghasemi V. et al.

FIG. 13. Dependency of ⟨u˜ θ ⟩zθ
0.0244ω −0.42 (b).

Phys. Fluids 28, 056603 (2016)

max

on E (a) and ω (b). Solid lines are the best fit line: 0.003 24E −0.46 (a) and

Figure 13(a) shows ⟨u˜θ ⟩zθ max for different Ekman numbers E ∈ 3 × 10−5, 2 × 10−4 , ε = 0.6,
and ω = 0.514. Dependency of ⟨u˜θ ⟩zθ max on E revealed E −0.46-scaling, which is weaker than E −0.5scaling predicted by (17). This deviation might be due to effect of the lids, nonlinearity of the regime
investigated, and/or conducting coarse numerical simulations as mentioned for the dependency

on ε.
Numerical data exhibit ω−0.42-scaling for the dependency of ⟨u˜θ ⟩zθ max on the libration frequency ω (Figure 13(b)), which is weaker than ω−0.5-scaling predicted by (17). General dependency
is physically understandable, since an increase in ω leads to a decrease of the Stokes boundary layer
thickness according to ∂r uθω ∝ δ−1 ∝ ω0.5 (uθω denotes the velocity profile of the Stokes boundary
layer (Sec. II C)), and thus to the smaller and weaker Görtler vortices.


′ ′
Figure 14 shows additionally radial profiles of ⟨u˜θ ⟩zθ (scaled by E −0.5) and u
r uθ zθ for different


′ ′
E. As E increases, ⟨u˜θ ⟩zθ
outside the Stokes boundary
decreases. However, slopes of u
ru
θ zθ

max

layer remain approximately constant with uncertainty of about 10% since ε is constant. As E increases, boundary layer velocity profile affects a larger fraction of the fluid volume since δ ∝ E 0.5,
however ∂r uθω ∝ δ−1 ∝ E −0.5 decreases. Thus, instability related to this primary (laminar) shear flow
weakens and leads to the lagged and less intense Görtler vortices. Moreover, by increasing E, the
azimuthal mean flow penetrates further into the bulk (Figure 14) due to an increase in δ. In fact, δ
gives wavelength of the oscillatory flow uθω and thus provides intrinsically a reference spatial scale
of the Görtler vortices. As for all other vortices, lifetime is increasing with size of the vortex.




′ u′
FIG. 14. Radial profiles of ⟨u˜ θ ⟩zθ × E 0.5 (a) and u
r θ

zθ

(b) for different Ekman numbers E, ε = 0.6, and ω = 0.514.


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Ghasemi V. et al.

Phys. Fluids 28, 056603 (2016)

It should be pointed out that here we only tried to understand behavior of the Görtler vortices and
generation of an azimuthal mean flow on the level of relation (16). Deeper investigation is beyond
the scope of this paper.

IV. SUMMARY AND CONCLUSION

Direct numerical simulation (DNS) has been performed for the outer and inner cylinder side wall
libration cases in a closed annulus in supercritical regime. We found generation of the Görtler vortices
in the rotationally unstable Stokes boundary layer, and their explosive propagation into the fluid bulk
due to effect of the Coriolis force. The Görtler vortices transport angular momentum from the boundary layer into the bulk and release it by mixing with the bulk flow. Here, a retrograde (prograde)
azimuthal mean flow was found in the bulk close to the librating outer (inner) cylinder side wall. The
responsible mechanism can be characterized as angular momentum pumping from the boundary layer
into the bulk by the Görtler vortices.
The azimuthal mean flow induced by the Görtler vortices has been compared with that generated
by the nonlinear effects in the oscillatory Ekman layer. Hereto a series of “coarse” numerical simulations (with coarse resolution in the azimuthal direction) has been conducted using the librational

boundary conditions listed in Table II. The results revealed that azimuthal mean flow in the bulk
generated by the Görtler vortices has similar magnitude to the one due to the nonlinearities in the
oscillatory Ekman boundary layer (in the case of librating lids).
The RANS equations derived based on the phase averages are used to investigate generation
mechanism of the azimuthal mean flow. Azimuthal component of the RANS equations revealed
a

proportionality between temporal derivative of basin-scale mean flow ∂⟨u¯θ ⟩zθ /∂t and ∂ ur′ uθ′ /∂r
zθ
(relation (12)) for unstable phase of the
cycle. Relation (12) exhibits that the sign of azimuthal
 libration

mean flow is given by the sign of ∂ ur′ uθ′ /∂r outside the Stokes boundary layer.
zθ
Scaling behavior of the azimuthal mean flow is explored using azimuthal component of the RANS
equations derived based on the long time average.
 analysis revealed a balance between radial
 Scale
′ u′
turbulent transport of the azimuthal momentum ∂ u
r θ zθ /∂r as a forcing term and radial dissipation
2
2
of the azimuthal mean flow E∂ ⟨u˜θ ⟩zθ /∂ r (relation (16)). This balance equation is confirmed by a
coarse numerical simulation for the axially periodic annulus. In a closed annulus, the mean radial
transport and Coriolis term need to be both considered in the balance equation (see (18)) due to
effect of the lids. Dependency of maximum amplitude of the azimuthal mean flow on the dimensionless parameters revealed E −0.46, ω−0.42, and ε 1.86-scaling behavior. Effect of the lids, nonlinearity of
the regime investigated, and/or conducting coarse numerical simulations seem to be the reasons of
deviations from the predicted scaling according to (16).

The mechanism discussed in this paper is applicable to a fluid flow over rotating curved surfaces.
According to the work of Chen and Lin (2002) and Zebib and Bottaro (1993), the rotational instability
condition can be fulfilled for the retrograde (prograde) flow over the rotating convex (concave) surfaces, causing continuous generation of the Görtler vortices. The vortices may generate a mean flow,
modifying the fluid flow in the bulk. Thus, we conjecture that it might explain asymmetry between
prograde and retrograde motions, in particular in the equatorial region (as reported by Griffiths, 2003).
Generation of the Görtler vortices in the equatorial region of a spherical shell has been reported
by Calkins et al. (2010) and Sauret et al. (2013). In this case azimuthal mean flows are usually
explained by the nonlinearities in the oscillatory Ekman layer. However, the boundary layer in the
equatorial region is nearly parallel to the rotation axis, implying that it is similar to the Stokes boundary layer. Therefore, it would be interesting to investigate the generation of azimuthal mean flow
by the Görtler vortices in a spherical shell.

ACKNOWLEDGMENTS

This work is part of the project “Mischung und Grundstromanregung durch propagierende
Trägheitswellen: Theorie, Experiment und Simulation,” financed by the German Science Foundation


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Phys. Fluids 28, 056603 (2016)

(DFG). The authors wish to thank Hans-Jakob Kaltenbach for providing a documented version of the
incompressible Navier–Stokes solver for axisymmetric diffuser flow. We also thank the anonymous
reviewers for their critical remarks that helped to improve the quality of the final version.
Batchelor, G. K., An Introduction to Fluid Dynamics (Cambridge University Press, 1967).
Bilson, M. and Bremhorst, K., “Direct numerical simulation of turbulent Taylor–Couette flow,” J. Fluid Mech. 579, 227–270
(2007).
Borcia, I. D., Ghasemi, A., and Harlander, U., “Inertial wave mode excitation in a rotating annulus with partially librating

boundaries,” Fluid Dyn. Res. 46, 041423 (2014).
Bretherton, F. P. and Turner, J. S., “On the mixing of angular momentum in a stirred rotating fluid,” J. Fluid Mech. 32, 449–464
(1968).
Busse, F. H., “Zonal flow induced by longitudinal librations of a rotating cylindrical cavity,” Physica D 240, 208 (2010).
Calkins, M. A., Noir, J., Eldredge, J., and Aurnou, J. M., “Axisymmetric simulations of libration-driven fluid dynamics in a
spherical shell geometry,” Phys. Fluids 22, 086602 (2010).
Celik, I. B., Ghia, U., and Roache, P. J., “Procedure for estimation and reporting of uncertainty due to discretization in CFD
applications,” J. Fluids Eng. 130(7), 078001 (2008).
Chen, C. T. and Lin, M. H., “Effect of rotation on Goertler vortices in the boundary layer flow on a curved surface,” Int. J.
Numer. Methods Fluids 40, 1327 (2002).
Choi, H., Moin, P., and Kim, J., “Direct numerical simulations of turbulent-flow over riblets,” J. Fluid Mech. 255, 503
(1993).
Czarny, O., Serre, E., Bontoux, P., and Lueptow, R. M., “Interaction between Ekman pumping and the centrifugal instability
in Taylor–Couette flow,” Phys. Fluids 15, 467–477 (2003).
Davidson, P. A., Turbulence in Rotating, Stratified and Electrically Conducting Fluids (Cambridge University Press, 2013).
Drazin, P. and Reid, W., Hydrodynamic Stability (Cambridge University Press, 1981).
Floryan, L. M., “Görtler instability of boundary layers over concave and convex walls,” Phys. Fluids 29, 2380 (1986).
Görtler, H., “Dreidimensionales zur Stabilitätstheorie laminarer Grenzschichten,” J. Appl. Math. Mech. 35(9-10), 362–363
(1955).
Greenspan, H. P. and Howard, L. N., “On a time-dependent motion of a rotating fluid,” J. Fluid Mech. 17, 385 (1963).
Griffiths, S., “The nonlinear evolution of zonally symmetric equatorial inertial instability,” J. Fluid Mech. 474, 245
(2003).
Kim, J. and Moin, P., “Application of a fractional step-method to incompressible Navier-Stokes equations,” J. Comput. Phys.
59, 308 (1985).
Klein, M., Seelig, T., Kurgansky, M. V., Borcia, I. D., Ghasemi, A., Will, A., Schaller, E., Egbers, C., and Harlander, U., “Wave
attractors and wave excitation in a liquid bounded by a frustum and a cylinder: Experiment, simulation and theory,” J. Fluid
Mech. 751, 255 (2014).
Kloosterziel, R. C., Carnevale, G. F., and Orlandi, P., “Inertial instability in rotating and stratified fluids: Barotropic vortices,”
J. Fluid Mech. 583, 379 (2007).
Kloosterziel, R. C. and van Heijst, G. J. F., “An experimental study of unstable barotropic vortices in a rotating fluid,” J. Fluid

Mech. 223, 1–24 (1991).
Le, H., “Direct numerical simulation of turbulent flow over a backward-facing step,” Ph.D. thesis, Department of Mechanical
Engineering Stanford University, 1994.
Lopez, J. M. and Marques, F., “Instabilities and inertial waves generated in a librating cylinder,” J. Fluid Mech. 687, 171
(2011).
Maas, L. R. M., “Wave focusing and ensuing mean flow due to symmetry breaking in rotating fluids,” J. Fluid Mech. 437, 13
(2001).
Manton, M. J., “A note on the mixing of angular momentum in a neutrally buoyant fluid,” Aust. J. Phys. 26, 607 (1973).
Marcus, P. S., “Simulation of Taylor–Couette flow. Part 1. Numerical methods and comparison with experiment,” J. Fluid
Mech. 146, 45 (1984).
Moin, P., “Large eddy simulation of backward-facing stop flow with application to coaxial jet combustors,” Air Force Office
of Scientific Research 2-DJA-420, 1995.
Morinishi, Y., Lund, T. S., Vasilyev, V. O., and Moin, P., “Fully conservative higher order finite difference schemes for incompressible flow,” J. Comput. Phys. 143, 90 (1998).
Moser, R. D. and Moin, P., “Direct numerical simulation of curved turbulent channel flow,” NASA TM-85974, 1984.
Neitzel, G. P. and Davis, S. H., “Centrifugal instabilities during spin-down to rest in finite cylinders. Numerical experiments,”
J. Fluid Mech. 102, 329 (1981).
Noir, J., Calkins, M. A., Cantwell, J., and Aurnou, J. M., “Experimental study of libration-driven zonal flows in a straight
cylinder,” Phys. Earth Planet. Inter. 182, 98106 (2010).
Noir, J., Hemmerlin, F., Wicht, J., Baca, S., and Aurnou, J., “An experimental and numerical study of librationally driven flow
in planetary cores and subsurface oceans,” Phys. Earth Planet. Inter. 173, 141–152 (2009).
Orlandi, P., Fluid Flow Phenomena—A Numerical Toolkit (Kluwer, 2000).
Lord Rayleigh, “On the dynamics of revolving fluids,” Proc. R. Soc. A 93, 148 (1916).
Saric, W. S., “Görtler vortices,” Annu. Rev. Fluid Mech. 26, 379 (1994).
Sauret, A., Cébron, D., and Le Bars, M., “Spontaneous generation of inertial waves from boundary turbulence in a librating
sphere,” J. Fluid Mech. 728, R5 (2013).
Sauret, A., Cébron, D., Le Bars, M., and Le Dizès, S., “Fluid flows in librating cylinder,” Phys. Fluids 24, 1–22 (2012).
Scorer, R. S., Report Entitled: Vorticity in Nature, Department of Mathematics, Imperial College, London, 1965.
Scorer, R. S., “Origin of cyclones,” Sci. J. 2, 46 (1966).

056603-23

Ghasemi V. et al.

Phys. Fluids 28, 056603 (2016)

Thompson, R. O. R. Y., “A mechanism for angular momentum mixing,” Geophys. Astrophys. Fluid Dyn. 12, 221
(1979).
Thompson, J. E., Warsi, Z. U. A., and Mastin, C. W., Numerical Grid Generation: Foundations and Applications (NorthHolland, New York, US, 1985).
Wang, C. Y., “Cylindrical tank of fluid oscillating about a steady rotation,” J. Fluid Mech. 41, 581 (1970).
Zebib, A. and Bottaro, A., “Goertler vortices with system rotation: Linear theory,” Phys. Fluids A 5, 1206 (1993).
Zhang, X., Boyer, D. L., and Fernando, H. J. S., “Turbulence-induced rectified flows in rotating fluids,” J. Fluid Mech. 350,
97 (1997).